If by compatible you mean that they find shrimp to be yummy, tasty, and fun to hunt, then yes! :D

I've kept betta and shrimp, and have a tank right now that was designated for a betta, but ended up with some red cherries/fire reds in it. I'll probably go ahead and add a betta anyway.

Know this: you need lots, and lots, and lots of cover for the shrimp. And, no amount of cover will be enough. The betta will probably eat some of your shrimp anyway, and give you a wide grin in thanks.

I have a 20 Long, 6 female bettas and lots of cover. I hope they won't be able to catch them all.

Depends on how many you add. I started with about 15 cherries, and I'm waiting for the colony to get much larger before adding a fish. I'd be worried about adding a small number to a tank that's already established.

You could take the fish out, re-scape, add the shrimp, and then add the fish back in. That would give the shrimp a fighting chance.

I have a betta in a 5 gal with 3 cherries. I started out with 5, gave them 24hrs to settle in and find all the cover & then added the shrimp. He chased them around as he came across them the first day or so (2 died, one after he chased her about and the other... idk) but now he swims right up to them, sniffs them and continues on. Sometimes he doesn't even sniff, he just swims on by. My tank is very heavily planted, btw. It all depends on your betta. I had a feeling that mine would be ok with them when he swam up to a pond snail, it moved and he swam away at tops speed and hid for the next hour lmao.

I would like to add that the terms "hiding spot" and "cover" means much smaller plants than your typical water wisteria, hornwort bacopa etc. you will need mosses and carpeting plants, and other extremely small hidig places etc...